Does a solution for Wumpus World with neural networks exist?
Yes! If you read ahead to the chapters in reinforcement learning in the same book, you'll see that the wompus world appears again there. Techniques like Q-learning can be used to solve it, and since Q-learning involves learning the shape of a function, a neural network can be employed as a function approximator.

The basic idea is to treat this problem as an input/output mapping (states -> actions), and to learn which actions produce the greatest rewards. 

Note however, that these approaches rely on trial and error. The logic based approach reasons about the rules of the game, and can play reasonably well right away. The learning approach will need to try and fail many times before playing well.